Fons the icosahedron (faces) does not provide a uniform spherical sampling in the sense that the spherical harmonics are orthogonal (well spotted Dave!). If you look for example at my thesis p. 167 you can observe that with such distribution the 3rd order (sampled) spherical harmonics are not orthogonal. I never understood exactly why (though I think Nicolas once told me something on that line), but I think it boils down to the definition of a uniform (or regular) sampling on a sphere. What does it actually mean?? or better, what do we mean by saying that? On the other hand, this is an issue if you use the hermitian transpose as an inverse, but the pseudo-inverse should solve the problem, I believe.
This is what I know about loudspeaker arrangement, but I have no idea how this relate to the energy vector thing... (I don't actually know how this gains are derived... I always see these thing from a different point of view, that is damping of the singular values of the matrix you are inverting) >> it turns out that the 3rd >>degree spherical harmonics are neither normalised nor orthogonal >>when summed over the set of directions corresponding to the faces >>of an icosahedron (but lower degrees are). This is confirmed by >>the set of singular values obtained when doing the pseudo-inverse. How do you judge the orthogonality of the Sph. Harm. from the Sing Values? I assume that if the harmonics were orthogonal, the S.V. would be all equal (apart from the null-space of the pseudo inverse - i.e. S.V.=0, if you allow me this lack of math. rigour...) I hope this helps. Greetings from SF Filippo >Message: 9 >Date: Thu, 4 Nov 2010 02:11:21 +0100 >From: f...@kokkinizita.net >Subject: Re: [Sursound] Help !! -- For AMB-decoding theory freaks only >To: Surround Sound discussion group <sursound@music.vt.edu> >Message-ID: <20101104011121.gd4...@zita2> >Content-Type: text/plain; charset=us-ascii > >On Wed, Nov 03, 2010 at 10:36:45PM +0000, d...@york.ac.uk wrote: > >> I'm still not sure what is bothering me about this but I _think_ >> it's something to do with the precise nature of the relationship >> between the symmetries in the icosahedron and the symmetries in the >> 3rd order spherical harmonics. The pictures you posted look like a >> spatial aliasing problem but, as you say, the array does seem to >> have a reasonable degree of oversampling, so... > >I think you are right about this. Unless some more of my code is >completely wrong and has been for years, it turns out that the 3rd >degree spherical harmonics are neither normalised nor orthogonal >when summed over the set of directions corresponding to the faces >of an icosahedron (but lower degrees are). This is confirmed by >the set of singular values obtained when doing the pseudo-inverse. > >So my conclusion so far is that the code I was testing is probably >OK, but my expectations were not. > >Ciao, > >-- >FA > >There are three of them, and Alleline. > > > >------------------------------ > >Message: 10 >Date: 04 Nov 2010 07:17:05 +0000 >From: d...@york.ac.uk >Subject: Re: [Sursound] Help !! -- For AMB-decoding theory freaks only >To: Surround Sound discussion group <sursound@music.vt.edu> >Message-ID: <prayer.1.3.2.1011040717050.23...@webmail4.york.ac.uk> >Content-Type: text/plain; format=flowed; charset=ISO-8859-1 > >On Nov 4 2010, f...@kokkinizita.net wrote: > >> it turns out that the 3rd >>degree spherical harmonics are neither normalised nor orthogonal >>when summed over the set of directions corresponding to the faces >>of an icosahedron (but lower degrees are). This is confirmed by >>the set of singular values obtained when doing the pseudo-inverse. >> > >That would do it! The interesting thing is, if I interpreted your earlier >emails right, that having _some_ portion of 3rd order is better than having >none at all. That suggests that it might be worth looking at treating >different third order components differently, as in Richard Furse's >"discarded harmonics" approach >(http://www.muse.demon.co.uk/ref/speakers.html) > >Dave > _______________________________________________ Sursound mailing list Sursound@music.vt.edu https://mail.music.vt.edu/mailman/listinfo/sursound
